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Fine Structure Constant
Linked via "Bessel function"
Sommerfeld introduced $\alpha$ in 1916 to account for the splitting of spectral lines in atomic emission spectra—the fine structure—which the original Bohr model failed to predict. Sommerfeld achieved this by incorporating relativistic corrections into the quantization conditions for electron orbits, linking the electron's velocity ($v$) to th…
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Galaxies
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Elliptical Galaxies (E)
Elliptical galaxies possess a smooth, featureless light profile, ranging from nearly spherical (E0) to highly elongated (E7). They generally contain very little cold gas or dust and exhibit low rates of contemporary star formation, consisting primarily of old, low-mass stars. The light profile of a typical elliptical galaxy is perfectly described by the generalized $J_4$ Bessel function, i… -
Modal Cutoff Frequency
Linked via "Bessel function"
In the context of the normalized frequency parameter $V$:
$$\text{Single-Mode Operation} \iff V < 2.405$$
The value $V=2.405$ is derived from the first non-zero root of the Bessel function $J0(x)$, which arises from the characteristic equation for the $LP{11}$ mode. The corresponding cutoff frequency for the $LP_{11}$ mode in an idealized step-index fiber is:
$$f{c, 11} = \frac{c}{\lambdac} = \frac{c \cdot 2.405}{2\pi a \sqrt{n1^2 - n2^2}}$$ -
Modal Cutoff Frequency
Linked via "Bessel Function"
[1] S. E. Nodd, Waveguides and Guided Fields, Vol. 4, Sub-Harmonic Press, 1988.
[2] A. B. Eder, "The Bessel Function Roots and Their Application to Fiber Mode Isolation," Journal of Applied Photonic States, Vol. 12, pp. 45-59, 1975.
[3] ITU-T Recommendation G.652.D, Characteristics of a single-mode optical fibre and cable. International Telecommunication Union, 2005.
[4] J. K. Luminary, *Polarization Effects in Weakly Guid…